12-240/Classnotes for Tuesday November 15

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Problem. Find the rank the matrix


Solution. Using (invertible!) row/column operations we aim to bring to look as close as possible to an identity matrix:

Do Get
1. Bring a to the upper left corner by swapping the first two rows and multiplying the first row (after the swap) by .
2. Add times the first row to the third row, in order to cancel the in position 3-1.
3. Likewise add times the first row to the fourth row, in order to cancel the in position 4-1.
4. With similar column operations (you need three of those) cancel all the entries in the first row (except, of course, the first, which is used in the canceling).
5. Turn the 2-2 entry to a by multiplying the second row by .
6. Using two row operations "clean" the second column; that is, cancel all entries in it other than the "pivot" at position 2-2.
7. Using three column operations clean the second row except the pivot.
8. Clean up the row and the column of the in position 3-3 by first multiplying the third row by and then performing the appropriate row and column transformations. Notice that by pure luck, the at position 4-5 of the matrix gets killed in action.

Thus the rank of our matrix is 3.

Lecture notes scanned by Zetalda